Denormalize results of curve fit on normalized data

Question

I am fitting an exponential decay function with lsqvurcefit in Matlab. To do this I first normalize my data because they differ several orders of magnitude. However Im not sure how to denormalize my fitted parameters.

My fitting model is s = O + A * exp(-t/T) where t and s are known and t is in the order of 10^-3 and s in the order of 10^5. So I subtract from them their mean and divide them by their standarddeviation. My goal is to find the best A, O and T that at the given times t will result most near s. However I dont know how to denormalize my resulting A O and T.

Might somebody know how to do this? I only found this question on SO about normalisation, but does not really address the same problem.

Answer 1

When you normalize, you must record the means and standard deviations for each of your featuers. Then you can easily use those values to denormalize.

e.g.

A = [1 4 7 2 9]';
B = 100 475 989 177 399]';

So you could just normalize right away:

An = (A - mean(A)) / std(A)

but then you can't get back to the original A. So first save the means and stds.

Am = mean(A); Bm = mean(B);
As = std(A);  Bs = std(B);
An = (A - Am)/As;
Bn = (B - Bm)/Bs;

now do whatever processing you want and then to denormalize:

Ad = An*As + Am;
Bd = Bn*Bs + Bm;

I'm sure you can see that that's going to be an issue if you have a lot of features (i.e. you have to type code out for each feature, what a mission!) so lets assume your data is arranged as a matrix, data, where each sample is a row and each column is a feature. Now you can do it like this:

data = [A, B]

means = mean(data);
stds = std(data);

datanorm = bsxfun(@rdivide, bsxfun(@minus, data, means), stds);

%// Do processing on datanorm

datadenorm = bsxfun(@plus, bsxfun(@times, datanorm, stds), means);

EDIT:

After you have fit your model parameters (A,O and T) using normalized t and f then your model will expect normalized inputs and produce normalized outputs. So to use it you should first normalize t and then denormalize f.

So to find a new f by running the model on a normalized new t. So f(tn) where tn = (t - tm)/ts and tm is the mean of your training (or fitting) t set and ts the std. Then to get your correct magnitude f you must denormalize only f, so the full solution would be

 f(tn)*fs + fm

So once again, all you need to do is save the mean and std you used to normalize.